Topic
Monodromy theorem
About: Monodromy theorem is a research topic. Over the lifetime, 373 publications have been published within this topic receiving 7614 citations.
Papers published on a yearly basis
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Papers
19 Jan 2010
TL;DR: In this paper, a general analytic continuation of three-dimensional Chern-Simons theory from Lorentzian to Euclidean signature is proposed, which can be carried out by rotating the integration cycle of the Feynman path integral.
Abstract: The title of this article refers to analytic continuation of three-dimensional Chern-Simons gauge theory away from integer values of the usual coupling parameter k, to explore questions such as the volume conjecture, or analytic continuation of three-dimensional quantum gravity (to the extent that it can be described by gauge theory) from Lorentzian to Euclidean signature. Such analytic continuation can be carried out by rotating the integration cycle of the Feynman path integral. Morse theory or Picard-Lefschetz theory gives a natural framework for describing the appropriate integration cycles. An important part of the analysis involves flow equations that turn out to have a surprising four-dimensional symmetry. After developing a general framework, we describe some specific examples (involving the trefoil and figure-eight knots in S^3). We also find that the space of possible integration cycles for Chern-Simons theory can be interpreted as the 'physical Hilbert space' of a twisted version of N=4 super Yang-Mills theory in four dimensions.
524 citations
TL;DR: In this paper, an algebraic description of the Picard-Lefschetz-monodromy of a singularity is given, and it is shown that the eigenvalues of the monodromy are roots of unity.
Abstract: J. Milnor recently introduced the local Picard-Lefschetz-monodromy of an isolated singularity of a hypersurface. This is an important tool in the investigation of the topology of singularities. The monodromy is an action on a certain cohomology group and is defined in topological terms. In this paper we find an algebraic description of the monodromy. We construct by algebraic methods a regular singular ordinary linear differential operator, such that the monodromy of this singular operator coincides with the Picard-Lefschetz monodromy. As an application we prove that the eigenvalues of the monodromy are roots of unity. Our treatment is close in spirit to Grothendiecks theory of the Gauβ-Manin-connection.
412 citations
TL;DR: In this article, it was shown that there exist extremal domains D 0 with minimal condenser capacity, and that these domains are uniquely determined up to a boundary set of capacity zero.
Abstract: In this paper we investigate the following two extremal problems: A) Let F be a continuum in the extended complex plane that does not divide and let f(z) be a function analytic on F By D we denote domains in such that f(z) has a single-valued analytic continuation in D. Does there exist a domain D 0 with minimal condenser capacity B) Let f(z) be a function analytic in a neighborhood of infinity. By D we denote domains in , such that f{z) has a single-valued analytic continuation in D. Does there exist a domain D 0 with minimal logarithmic capacity It is proved that there exist extremal domains D 0 in both problems. In a second part of the paper it will be shown that these domains are uniquely determined up to a boundary set of capacity zero.
134 citations
Posted Content•
TL;DR: In this article, a canonical filtration for locally free sheaves on an open p-adic annulus equipped with a Frobenius structure is presented, and a conjecture of Grothendieck's local monodromy theorem is derived.
Abstract: We produce a canonical filtration for locally free sheaves on an open p-adic annulus equipped with a Frobenius structure. Using this filtration, we deduce a conjecture of Crew on p-adic differential equations, analogous to Grothendieck's local monodromy theorem (also a consequence of results of Andre and of Mebkhout). Namely, given a finite locally free sheaf on an open p-adic annulus with a connection and a compatible Frobenius structure, the corresponding module admits a basis over a finite cover of the annulus on which the connection acts via a nilpotent matrix.
Note: this preprint improves on results from our previous preprints math.AG/0102173, math.AG/0105244, math.AG/0106192, math.AG/0106193 but does not explicitly invoke any results from these preprints.
128 citations
Posted Content•
TL;DR: In this paper, the authors employ a mapping between the analytic continuation problem and a system of interacting classical fields, and show that the maximum entropy method is a special limit of the stochastic analytic continuation method introduced by Sandvik.
Abstract: The maximum entropy method is shown to be a special limit of the stochastic analytic continuation method introduced by Sandvik [Phys. Rev. B 57, 10287 (1998)]. We employ a mapping between the analytic continuation problem and a system of interacting classical fields. The Hamiltonian of this system is chosen such that the determination of its ground state field configuration corresponds to an unregularized inversion of the analytic continuation input data. The regularization is effected by performing a thermal average over the field configurations at a small fictitious temperature using Monte Carlo sampling. We prove that the maximum entropy method, the currently accepted state of the art, is simply the mean field limit of this fully dynamical procedure. We also describe a technical innovation: we suggest that a parallel tempering algorithm leads to better traversal of the phase space and makes it easy to identify the critical value of the regularization temperature.
120 citations
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| Year | Papers |
|---|---|
| 2021 | 1 |
| 2020 | 3 |
| 2018 | 3 |
| 2017 | 10 |
| 2016 | 14 |
| 2015 | 12 |